TL;DR
This paper introduces the concept of dimension as a geometric invariant for classifying martingales and extends the definition of Brownian motions to higher dimensions.
Contribution
It defines the dimension of martingales based on the rank of the integrand and generalizes Brownian motion to higher-dimensional settings.
Findings
Dimension serves as a geometric invariant for martingales.
Martingales can be classified by their dimension.
Higher-dimensional Brownian motions are formally defined.
Abstract
A martingale \int H.dZ is defined as having Dimension k if H has rank k almost surely, almost all t. Dimension can be used as a geometric invariant to classify and study martingales. We also define general Brownian motions in higher dimensions.
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Taxonomy
TopicsStochastic processes and financial applications · Insurance, Mortality, Demography, Risk Management